A Novel Technique of Neural-Network Extension for Estimating Multi-dimensional Integrals of an Arbitrary Function

In this paper, a novel technique of neural-network extension is proposed to estimate multi-dimensional integrals of an arbitrary function. Initially, the nonlinear mapping between integral variables and the integrand (i.e., the function to be integrated) is constructed by an RBF-NN (Radial Basis Function Neural Network). This RBF-NN has one node in the output layer representing the integrand and several nodes in the input layer representing integral variables. There also exist some nodes in the hidden layer for nonlinear transformation. After the RBF-NN is well trained, the output, i.e., the integrand, can be generalized by weights and Gaussian bases within the neural network. Therefore, the original integration can be transformed into multi-dimensional integrals on different Gaussian bases and the final results can be achieved by look-up tables of mathematical handbooks. Numerical simulation shows that our neural-network based technique is feasible and accurate. It should be noted that no numerical integration procedure is required by the proposed technique. With the use of neural network, only discrete sampling points for the integrand are required. In other words, one does not need to know the overall characteristic of the integrand. This study can be applied to many engineering problems with complicated integrals.